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A metric space X is isometric to a metric space Y if there is a bijection f between X and Y that preserves distances. That is, d(a,b)=d(f(a),f(b)). In the context of ...

A point x^* which is mapped to itself under a map G, so that x^*=G(x^*). Such points are sometimes also called invariant points or fixed elements (Woods 1961). Stable fixed ...

The Ricci curvature tensor, also simply known as the Ricci tensor (Parker and Christensen 1994), is defined by R_(mukappa)=R^lambda_(mulambdakappa), where ...

Every closed three-manifold with finite fundamental group has a metric of constant positive scalar curvature, and hence is homeomorphic to a quotient S^3/Gamma, where Gamma ...

The tangent plane to a surface at a point p is the tangent space at p (after translating to the origin). The elements of the tangent space are called tangent vectors, and ...

The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism, it is a map. In particular, an Anosov map is a C^1 ...

A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on ...

There are no fewer than two closely related but somewhat different notions of gerbe in mathematics. For a fixed topological space X, a gerbe on X can refer to a stack of ...

The operator partial^_ is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative d takes a function and yields a one-form. It decomposes ...

A point where a stable and an unstable separatrix (invariant manifold) from the same fixed point or same family intersect. Therefore, the limits lim_(k->infty)f^k(X) and ...